Magnetic Resonance and Bloch Equations

This Demonstration visualizes the dynamics in the process of magnetic resonance in which the macroscopic magnetization of an ensemble of paramagnetic particles is exposed to the common action of a static magnetic field and a weak magnetic field () that rotates with a frequency around . The motion of is governed by the Bloch equations. The effect of on is maximized when the rotational frequency is equal to the Larmor free precession frequency (i.e. detuning ).

Many interesting cases are registered as bookmarks, which you can activate by clicking the small cross at the upper-right corner. For example:

• Free Larmor precession, which occurs for , so that magnetization precesses around at the Larmor frequency.

• A -pulse, for which rotates at the Larmor frequency (detuning =0) and for which is switched on for a time duration , such that . As a result, the magnetization is flipped by into the - plane.

• A -pulse, for which rotates at the Larmor frequency (detuning =0) and for which is switched on for a time duration , such that . As a result, the magnetization is flipped by to the direction.

• Magnetic resonance with relaxation, for which the magnetization reaches a steady state. You can play with the value of the frequency detuning . If or , the effect of is small and the steady state is close to the equilibrium magnetization along . However, when , the effect of is important and the steady state is reached far from the axis.

• Adiabatic following occurs for , in which case the magnetization precesses rapidly around the magnetic field and therefore follows the direction of adiabatically.

THINGS TO TRY

SNAPSHOTS

DETAILS

The time evolution of the magnetization of an ensemble of magnetic moments in a magnetic field is described by the Bloch equations,

,

where is the equilibrium -component of , when all fields are 0; and are called the longitudinal and transverse relaxation rates, respectively. The total magnetic field is the vector sum of a static field along and a field rotating in the - plane,

.

Inserting into the Bloch equations yields

,

where is the free Larmor precession frequency around and is called the Rabi frequency, which characterizes the magnitude of , the strength of the interaction of the rotating field with the magnetization.

In the Demonstration, we set , so all frequencies and relaxation rates are expressed in units of . The time unit is therefore and the total time is equivalent to the number of Larmor cycles.